Topics in Algebraic Geometry

††††††††††† †††Math 647, Spring 2013

 


Course description:

The main prerequisite is a one year introductory course on algebraic geometry (e.g. chapters II-IV in Hartshorne's textbook).

 

The aim of the course is to study more advanced topics, and this will be organized as follows. The main goal is to study moduli spaces of smooth curves and their compactifications, for this we will have to study the following topics: stable curves,Hilbert schemes, Grothendieck topologies, algebraic spaces and stacks, moduli spaces of stable n-pointed curves, and stable reduction theorem. As the main application of the developed theory we will prove in the end of the course de Jong's theorems on semi-stable families and desingularization of varieties by alterations.

 

There is no final exam. Instead of this, students that want to take the course for credit will give a talk in the end of the course.

Texts:

DM, the article by Deligne-Mumford on irreducibility of moduli spaces.

AO, lecture notes by Abramovich-Oort on de Jongís results and related stuff.

Lecture notes, where I wrote down lectures of an analogous course given at University of Pennsylvania. (You can ignore the division to lectures since our 3-hour meetings will cover more material.) Any remark on a mistake/inaccuracy will be very welcome (and Iím sure that there are some).